Advertisements
Advertisements
प्रश्न
Show that `cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
उत्तर
We have to prove that `cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
L.H.S = `cot(7 1^circ/2)`
= `(cos(7 1^circ/2))/(sin(7 1^circ/2))`
To find `costheta/sintheta`, multiply numerator and denominator by 2 cos θ
Let θ = `71/2^circ`
2θ = 15°
`(2cos^2theta)/(2sin theta cos theta) = (1 + cos 2theta)/(sin 2theta)`
= `(1 + cos 15^circ)/(sin 15^circ)`
= `((1 + sqrt(3) + 1)/(2sqrt(2)))/((sqrt(3) - 1)/(2sqrt(2))`
= `(2sqrt(2) + sqrt(3) + 1)/(sqrt(3) - 1)`
Multiply numerator and denominator by `sqrt(3) + 1`
= `((2sqrt(2) + sqrt(3) + 1)(sqrt(3) + 1))/((sqrt(3) - 1)(sqrt(3) + 1))`
= `(2sqrt(2) + 3 + sqrt(3) + 1)/(3 - 1)`
= `(2sqrt(3) + 2sqrt(2) + 4)/2`
= `(2(sqrt(2) + sqrt(3) + sqrt(6) + 2))/2`
= `sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
= R.H.S
APPEARS IN
संबंधित प्रश्न
Find the values of `tan ((19pi)/3)`
Find the value of the trigonometric functions for the following:
sec θ = `13/5`, θ lies in the IV quadrant
If sin A = `3/5` and cos B = `9/41, 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of cos(A – B)
Find sin(x – y), given that sin x = `8/17` with 0 < x < `pi/2`, and cos y = `- 24/25`, x < y < `(3pi)/2`
Prove that cos(30° + x) = `(sqrt(3) cos x - sin x)/2`
Prove that sin(π + θ) = − sin θ.
Expand cos(A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = `pi/2`
Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)
If tan x = `"n"/("n" + 1)` and tan y = `1/(2"n" + 1)`, find tan(x + y)
Find the value of tan(α + β), given that cot α = `1/2`, α ∈ `(pi, (3pi)/2)` and sec β = `- 5/3` β ∈ `(pi/2, pi)`
Prove that (1 + tan 1°)(1 + tan 2°)(1 + tan 3°) ..... (1 + tan 44°) is a multiple of 4
Prove that (1 + sec 2θ)(1 + sec 4θ) ... (1 + sec 2nθ) = tan 2nθ
Express the following as a sum or difference
2 sin 10θ cos 2θ
Express the following as a sum or difference
sin 5θ sin 4θ
Express the following as a product
cos 65° + cos 15°
Show that sin 12° sin 48° sin 54° = `1/8`
Prove that `sin theta/2 sin (7theta)/2 + sin (3theta)/2 sin (11theta)/2` = sin 2θ sin 5θ
Prove that cos(30° – A) cos(30° + A) + cos(45° – A) cos(45° + A) = `cos 2"A" + 1/4`
If A + B + C = 180°, prove that sin2A + sin2B + sin2C = 2 + 2 cos A cos B cos C
Choose the correct alternative:
If `pi < 2theta < (3pi)/2`, then `sqrt(2 + sqrt(2 + 2cos4theta)` equals to
